Indices of complexity, entropy and fractal dimension

[DominiqueMakowski]

This is a summary of the progress of implementing a comprehensive framework for complexity analysis. The goal is too carefully reimplement all indices so that it's efficient, validated and tested.

Measures

Entropy

• Shannon entropy (neurokit2 == pyEntrp)
• Approximate entropy (neurokit2 == EntroPy != pyeeg ?= pyrem)
• corrected ApEn (cApEn) (paper)
• Sample entropy (neurokit2 == EntroPy == nolds == entro-py == pyEntrp == pyeeg ?= pyrem)
• Fuzzy entropy (neurokit2 == entro-py ?= matlab)
• multiscale entropy (MSE) (pyEntrp)
• Fuzzy MSE
• Composite multiscale entropy (CMSE) (pyEntrp)
• Fuzzy CMSE
• Refined composite multiscale entropy (RCMSE) (paper)
• Fuzzy RCMSE
• Multidimensional Approximate Entropy (MApEn) (paper)
• Generalized multiscale entropy (GMSE) (in C)
• Permutation entropy (pyEntrp, EntroPy)
• Multiscale permutation entropy (pyEntrp)
• conditional entropy (CE) (paper)
• distribution entropy (DistEn) (paper)
• Spectral entropy (EntroPy, pyrem)
• Singular value decomposition entropy (EntroPy, pyrem)
• Inherent fuzzy entropy (matlab)
• Rényi entropy (dit)
• Tsallis Entropy (dit)
• Entropy estimation via compression coding (PNG-Entropy by Zbili, 2020)
• ...

Fractal Dimension

• detrended fluctuation analysis (DFA) (neurokit2 == nolds, EntroPy, pyrem, fathon, dfa)
• multifractal detrended fluctuation analysis (MFDFA) (fathon, MFDA, PyActigraphy)
• correlation dimension (nolds, Corr_Dim, paper)
• Lyapunov exponent (nolds)
• Hurst exponent (nolds, pyrem)
• Generalized Hurst exponent (GenHust)
• Petrosian fractal dimension (EntroPy, pyrem)
• Katz fractal dimension (EntroPy)
• Higuchi fractal dimension (EntroPy, pyrem)
• Multiscale multifractal analysis (matlab)
• ...

Other

• Hjorth's parameters and complexity (pyrem)
• Lempel-Ziv complexity (EntroPy)
• Ficher information (pyrem)
• delay vector variance (DVV) (paper)
• Noise reduction (nolitsa)
• Kramers-Moyal Coefficients (KramersMoyal by @LRydin; see Prusseit, 2007 for EEG application)

Helpers

• Optimal Embedding Dimension
• Optimal Time Delay: here
• Chaotic Attractor: here
• Chaos Decision Tree Algorithm (matlab; Toker et al., 2020)

References

• Amarantidis, L. C., & Abásolo, D. (2019). Interpretation of entropy algorithms in the context of biomedical signal analysis and their application to EEG analysis in epilepsy. Entropy, 21(9), 840.
• Gao, Z., Cui, X., Wan, W., & Gu, Z. (2019). Recognition of Emotional States using Multiscale Information Analysis of High Frequency EEG Oscillations. Entropy, 21(6), 609.

[LRydin]

Small comment, I guess there is little point in separating detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MFDFA), as the latter should always be able to do DFA if you set the power/exponent =2 :).

[DominiqueMakowski]

Thanks for that very good point 😁 I'll leave it as is just to separate the implementations (as most of DFA implementations do not allow for MDFA ;)

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